Heuristics for Lipschitz equivalence

I'm studying for my finals in general topology and when I look at the definition of the Lipschitz equivalence of two metrics as:

Let $d(x,y)$ and $d'(x,y)$ be metrics on a non-empty set $X$. We define $d$ and $d'$ to be Lipschitz equivalent if there exists $\alpha, \beta > 0$ such that

$$\alpha d'(x,y) \leq d(x,y) \leq\beta d'(x,y) .$$

I noticed that there is a similarity between this definition and the $\Theta-$ notation for the asymptotics of a function. I'm wondering if I could possibly use the definition of Lipschitz eqivalence as

$$\lim_{x,y\to \infty} \frac{d(x,y)}{d'(x,y)}=v$$

where $0<v<\infty$ as an equivalence formulation of Lipschitz equivalence? Also, is there a version of L'Hopital's rule for multivariate functions?

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For your last question, see L'Hôpital in several variables. – Dave L. Renfro May 30 '14 at 14:58

There is indeed some similarity. But, in the $\Theta$-notion we have the idea that this approximation only holds for sufficiently large $x$, while here we have that the inequality holds for all pairs of points, so it's uniform. Also, it's unclear what the equivalent formulation would be of the notion of limit towards infinity, e.g. What if a metric space is bounded?

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You are on to something here, although the guess is incorrect. Suppose the Lipschitz equivalence fails: that is, either $\alpha$ or $\beta$ fails to exist. For definiteness, consider the case then $\alpha$ fails to exist; the other case amounts to exchanging $d$ and $d'$. Since there is no $\alpha$, there exist sequences $x_n$, $y_n$ such that $$\lim_{n\to\infty} \frac{d(x_n,y_n)}{d'(x_n,y_n)} = 0 \tag{1}$$ How can (1) happen? Maybe $d'$ is large, or maybe $d$ is small. So, passing to a suitable subsequence, we obtain either $$\lim_{n\to\infty} \frac{d(x_n,y_n)}{d'(x_n,y_n)} = 0, \quad \lim_{n} d'(x_n,y_n) = \infty$$ or $$\lim_{n\to\infty} \frac{d(x_n,y_n)}{d'(x_n,y_n)} = 0, \quad \lim_{n} d(x_n,y_n) = 0$$ What this says is that if the Lipschitz equivalence fails, it fails on an extreme scale for one of the metrics: either very large, or very small scale.

The above observations are actually useful, especially in two special cases:

1. the space is bounded (both $d,d'$ are bounded from above)
2. the space is discrete (nonzero distances $d,d'$ are bounded from below)

In the first case we are only concerned with small scale behavior; in the second case only with large scale behavior.

Also, is there a version of L'Hopital's rule for multivariate functions?

Certainly not on general metric spaces, where we don't have derivatives. On $\mathbb R^n$? Can't say for sure. You could ask this as a separate question.

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